Tip:
Highlight text to annotate it
X
So last time we saw that we could actually test for type1 zeno and the way we could
test this was to see whether or not the gradient dg, dx transpose points in the
same direction as f2, in the opposite direction of f1. So if I have that, then I
do have type one zeno. And we came up with a little probe here to check whether or
not, this is indeed happening. So, the first thing we do is we check LF1G
negative, and LF. two G positive. Where LFG was this thing we called the lead
derivative. Which was simply just fancy speak for DG, DX times, in this case, F1
of X. That's all it means. So this is the probe that checks whether or not sliding
occurs. Today, I want to talk about what's called regularizations, which is. Code for
okay, let's say we have that, we have sliding, how do we compute how we slide?
And it turns out that this is going to be very important when we move on to robotics
because this occurs quite a bit when you actually start running robotic navigation
systems. So today's lecture is all about How to be actually slide along this
surface.Well, lets actually try to be a little careful in . So if I'm sliding
here, what is characterizing that motion well, i know that g(x)=0 which means that
if i take the time, the relative of g, we respect to t Then it should be zero
because g is not going to change when I'm sliding because I'm staying at g=0.
So one thing we need to do is say that dg/dt=0.
Well, let's see where that takes us, dg/dt It is equal to dgdx*xdot.
So now what is x dot along here. Well, what I am going to postulate is that it's
a combination of f2 and f1. In fact let's say that is equal to sigma1*f1+sigma2+f2.
So it's a convex combination of, of these two Vector fields. So if I do that, I get
instead of x dots, I get this thing here sigma one f1 plus sigma two f2. Well,
since these are scalars, I can actually pull them out here, which means that I can
write this expression that's is a little bit messy looking like sigma one times the
lead derivative of g along f1 plus sigma two times the lead derivative of g alo ng
f2. So that's what the time derivative of g actually is. But I want that to be equal
to zero. This needs to be equal to zero because
that was our condition. So if I put this thing equal zero then I can solve for
instance for sigma two and I get. This expression right here. So I know how two
is going to depend on one. Well that's a good start. We also know
that both need to be positive because I'm not allowed to always start flowing
backwards and they also should sum up to one because otherwise they can go super
fast along this direction which I typically don't want. I want it to respect
the dynamics. So. I have additional constraint too. So the sigma needs to be
positive and they need to sum up to one. Well. let's see how we can actually solve
it. So, let's go back to our old friend, the example. And again this example. Looks
like this, where, you know what? I'm sliding down with slope -one.
Here's x and here's time. And when I hit the switching surface, I stay with x=0 for
the duration. Okay, what is the switching surface, first of all? Well, it's x=0, so
g(x) is simply x=0. Okay, let's compute some of these Lee
derivatives. So, lf1g. It's, it's the derivative of g, with respect to x. So,
the derivative of that with respect to x, is simply one.
F1 is simply negative one. So, the Lee derivative is negative one.
Which is one of the things we needed for zeno. We needed, or for sliding. We needed
this to be negative for, type one zeno to occur. That's the first. Well let's do the
same for f two. Sorry this is an f two right here,
apologize about that. l f two g well it's a dgdx which equals the derivative of that
with respect to x which is one, and f two which is plus one.
Right, so I get one*1, which is equal to one, which is positive, which was the
other condition for having Type one Zeno. So we know that we actually have Type one
Zeno, or sliding, here. okay. So now, let's try figuring out what, what the, The
sigmas should actually be. Well, we had a formula for sigma 2.. It is equal to
-sigma1 and these two Lee derivatives divided by eachother. Well, this w as -one
and this was +one so sigma two is simple equal to sigma one.
Rather simple. . But what, what are they? Well, recall that the need to sum up to
one. Which means that they both have to be a
half. What this actually means, then, is that we can compute what the induced mode,
sliding mode, actually is. Well, I take sigma one * f1, which is half of f1. And
then i take sigma2 times F2 which is half of F2, well F1 is negative one and F2 is
plus one so i get -0.5 + 0.5 and that's equal to zero.
So I know that my induced mode, in this case, is x dot = zero, which is exactly
what we wanted. because remember this picture that we've drawn over and over
again. We want to start here, get down there, and then keep staying at zero.
And the math turns out to work out, in this case. Now, let's find the induced
mode in general. I know that the general formula for sigma2 and I, I want to point
out that this is not a formula any one in their right mind should memorize, but we
should know where it comes from and be able to use it when we need to. But, here
is the general formula for the, for the induced mode. Well, we also have that
sigma one plus sigma two, is equals to one, which means that I can take sigma two
here, and plug it in, because this is what sigma two is. So now I have an expression
on sigma one so if I solve that, I acutally know what sigma one is. I
encourage you to go through the math yourself, it's a little bit of a mouth
full, but what its, what it tells me is that I compute sigma one, and I can
compute sigma two, just as well. And, if I now sum all of this up together, I get
this expression inside the pink box there. And I recognize again, this is a little
bit of a mouthful, but what this tells me is Exactly what the induced mode is. So I
am going to put a sweet heart around this thing, not because it's particularly
pretty but because it's systematic and it tell us how to actually find the sliding
mode controller or the induced mode which means that we know Exactly, not only when
type one zeno occurs. But how to progress beyond it. So, let's figur e it out. If I
have, this is my hybrid system. And, if this is a type one zeno hybrid automaton.
How do I regularize it? What do I do to add this extra sliding mode? Well, this is
what I do. Now, let's parse this, even though it looks a little bit like a
mouthful, let's figure out what it means. So let's say I'm here, g is positive. G is
positive and all is, all is well. Then, if g becomes zero and the sliding condition
is satisfied. Then I move in to what we just computed here which is the induce
mode. And again it's a little bit of a mouth full but we will see later on that
in robotics we have no choice but to actually use this. So then I'm going to
use this mode until if g becomes positive I jump back to f1. If g becomes negative I
jump to f2. And of course we don't always have Type one Zeno some switches are nice
if I simply end up g negative, I jump directly from mode one to mode two and
vice versa. So, this is how you take a hybrid system and make it immune to the
Nauseous and bad effects of Type one Zeno. And the nice thing is that this is
completely general, and we don't ever again have to worry about Zeno. Type one
Zeno, Type two Zeno we already said them about. Having said that this actually
brings me to the summary of this entire model and hybrid systems. So, what do we
have, we have models, we have very rich models which are the hybrid models. We
have something what we call Stability Awareness or just We're aware of the fact
that, just because the submodules, or the submodes are stable themselves doesn't
mean that the hybrid system is stable. And we need to be aware of it, test for it. We
also have seen zeno as another awkward hybridization that occurs. Or an awkward
phenomenon that can occur when you go hybrid. we have two classes of zeno. one,
which is type two, which is the bouncing ball. Infinitely many switch, many
switches in finite, but not, not zero time. That's bad and scary, and we can't
do anything about it except, look out for it. But then, we have type one.
And we now know, not only how to check for it. But how to get around it using these
things that I call regularizations or the induced sliding mode and with that if I
had, you know a confetti, I would toss it up into the air because this ends the
massive part and the pre-robotics part really of this course. So what we're going
to do in the next module is go back to robotics. Apply, all of our new and
awesome tools, and see, how we can unleash them to actually make mobile ro, robots do
cool things in the real world. Well done.